Exercise 11, § 5

Let $g$ be a Lie algebra, $\rho$ a finite-dimensional representation of $g$ and $E$ the associative endomorphism algebra generated by $1$ and $\rho(g)$. Show that the largest nilpotency ideal $n$ of $\rho$ is equal to the set $n'$ of $x \in g$ such that $\mathrm{Tr}(\rho(x)u) = 0$ for all $u \in E$. (To show that $n' \subset n$, show that $n'$ is an ideal and that for all $x \in n'$ the semi-simple component of $\rho(x)$ is zero, by noting that $\mathrm{Tr}((\rho(x))^n) = 0$ for every integer $n > 0$.)